Visual models of identities

The parts (4x^2) and (9y^2) give side parts (2x) and (3y). Exam tip: identify the side parts from both square terms first.

After subtracting two strips, (b^2) must be added back. Exam tip: watch the overlapping part carefully.

The difference of squares changes into ((x-b)(x+b)). Exam tip: the rectangle formed by cutting has difference and sum as sides.

This is product of sum and difference, so ((2m)^2-5^2=4m^2-25). Exam tip: square the whole (2m).

The total middle term is (18p), so each equal rectangle is (9p). Exam tip: split (2ab) into two equal rectangles.

The square has side \(3a-2b\), so its area is \((3a-2b)^2\). The middle term of a square of the form \((u+v)^2\) is \(2uv\). Here, the two parts are \(u=3a\) and \(v=-2b\). Therefore the middle term is \(2(3a)(-2b)=-12ab\). The negative sign comes from the negative second part of the side.

Thus option C is correct. The complete expansion is \((3a-2b)^2=9a^2-12ab+4b^2\). In a visual area model, the two side rectangles have areas \((3a)(-2b)=-6ab\) each; together they give \(-12ab\). Options A and B lose the negative sign or use an incorrect coefficient, while D has the wrong magnitude as well as the wrong sign.

The small corner area is (4\cdot 7=28). Exam tip: identify the product of constant parts separately.

The two \(xy\) rectangles have total area \(2xy\). All four tiles form a square of side \(x+y\), so its area is \((x+y)^2\). Option B has a negative middle term. Exam tip: add the tile areas before identifying the identity.

The rectangle sides are (a+b=15) and (a-b=7), so (a=11) and (b=4). Exam tip: find the two numbers from sum and difference.

For a square with side \(2x+3\), the area is \((2x+3)^2\). In the area model, the two mixed rectangles are formed by multiplying the variable part \(2x\) by the constant part \(3\). Each rectangle has area \((2x)(3)=6x\). Since there are two such rectangles, their combined area is \(6x+6x=12x\).

The same result follows from the identity \((u+v)^2=u^2+2uv+v^2\), with \(u=2x\) and \(v=3\). The mixed term is \(2(2x)(3)=12x\), so option C is correct. Option A gives only one rectangle, B confuses the constant with the total, and D is the area of the variable square alone. The phrase “two rectangles” means the mixed parts must be added together.

Removing two \(a\times b\) strips gives \(a^2-2ab\). Their common \(b^2\) corner was removed twice, so add it once back. The remaining square has side \(a-b\). Exam tip: add back the overlap in area models.

The side parts are (3x) and (4), so the subtraction square is ((3x-4)^2). Exam tip: identify the negative middle sign.

The numbers (4) and (5) have product (20) and sum (9). Exam tip: check corner product and middle sum together.

The combined middle area is (2\cdot 5p\cdot -q=-10pq). Exam tip: write subtraction in the visual model as a negative term.

The total area is (a^2+2ab+b^2=85+84=169). Exam tip: the sum of all parts gives total area.

When a square of side \(a+b\) is partitioned, it contains squares of areas \(a^2\) and \(b^2\). The remaining two rectangles each measure \(a\times b\), so their total area is \(ab+ab=2ab\). Exam tip: count both rectangles.

(16a^2=(4a)^2) and (25b^2=(5b)^2), and the middle term is (40ab). Exam tip: add the roots of both squares.

The four parts include like terms (15x) and (2x), whose sum is (17x). Exam tip: combine like terms.

In ((2x-3)^2), the middle term (2\cdot 2x\cdot 3=12x) is negative. Exam tip: square the coefficient of the first term.

(a^2-b^2=(a-b)(a+b)), so (96=6(a+b)). Exam tip: divide to find the other side of the rectangle.